chapter 4 – frequent pattern mining
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Chapter 4 – Frequent Pattern Mining. Shuaiqiang Wang ( 王帅强 ) School of Computer Science and Technology Shandong University of Finance and Economics Homepage: http://alpha.sdufe.edu.cn/swang/ The ALPHA Lab: http://alpha.sdufe.edu.cn/ [email protected]. Outline. Basic Concepts - PowerPoint PPT PresentationTRANSCRIPT
Chapter 4 – Frequent Pattern Mining
Shuaiqiang Wang (王帅强 )School of Computer Science and Technology
Shandong University of Finance and EconomicsHomepage: http://alpha.sdufe.edu.cn/swang/
The ALPHA Lab: http://alpha.sdufe.edu.cn/[email protected]
2
Outline
• Basic Concepts
• The Apriori Algorithm
• The FP-Growth Algorithm
3
What Is Frequent Pattern Analysis?
• Frequent pattern: a pattern (a set of items, subsequences, substructures, etc.)
that occurs frequently in a data set
• First proposed by Agrawal, Imielinski, and Swami [AIS93] in the context of
frequent itemsets and association rule mining
• Motivation: Finding inherent regularities in data
– What products were often purchased together?— Beer and diapers?!
– What are the subsequent purchases after buying a PC?
– What kinds of DNA are sensitive to this new drug?
– Can we automatically classify web documents?
• Applications
– Basket data analysis, cross-marketing, catalog design, sale campaign
analysis, Web log (click stream) analysis, and DNA sequence analysis.
4
Why Is Freq. Pattern Mining Important?
• Freq. pattern: An intrinsic and important property of datasets • Foundation for many essential data mining tasks
– Association, correlation, and causality analysis– Sequential, structural (e.g., sub-graph) patterns– Pattern analysis in spatiotemporal, multimedia, time-series,
and stream data – Classification: discriminative, frequent pattern analysis– Cluster analysis: frequent pattern-based clustering– Data warehousing: iceberg cube and cube-gradient – Semantic data compression: fascicles– Broad applications
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Basic Concepts: Frequent Patterns
• itemset: A set of one or more items• k-itemset X = {x1, …, xk}• (absolute) support, or, support
count of X: Frequency or occurrence of an itemset X
• (relative) support, s, is the fraction of transactions that contains X (i.e., the probability that a transaction contains X)
• An itemset X is frequent if X’s support is no less than a minsup threshold
Customerbuys diaper
Customerbuys both
Customerbuys beer
Tid Items bought
10 Beer, Nuts, Diaper
20 Beer, Coffee, Diaper
30 Beer, Diaper, Eggs
40 Nuts, Eggs, Milk
50 Nuts, Coffee, Diaper, Eggs, Milk
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Basic Concepts: Association Rules
• Find all the rules X Y with minimum support and confidence– support, s, probability that a
transaction contains X Y– confidence, c, conditional
probability that a transaction having X also contains Y
Let minsup = 50%, minconf = 50%Freq. Pat.: Beer:3, Nuts:3, Diaper:4, Eggs:3, {Beer,
Diaper}:3
Customerbuys diaper
Customerbuys both
Customerbuys beer
Nuts, Eggs, Milk40Nuts, Coffee, Diaper, Eggs, Milk50
Beer, Diaper, Eggs30
Beer, Coffee, Diaper20
Beer, Nuts, Diaper10
Items boughtTid
Association rules: (many more!) Beer Diaper (60%, 100%) Diaper Beer (60%, 75%)
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Closed Patterns and Max-Patterns• A long pattern contains a combinatorial number of sub-
patterns, e.g., {a1, …, a100} contains (1001) + (100
2) + … + (11
00
00) =
2100 – 1 = 1.27*1030 sub-patterns!• Solution: Mine closed patterns and max-patterns instead• An itemset X is closed if X is frequent and there exists no super-
pattern Y כ X, with the same support as X (proposed by Pasquier, et al. @ ICDT’99)
• An itemset X is a max-pattern if X is frequent and there exists no frequent super-pattern Y כ X (proposed by Bayardo @ SIGMOD’98)
• Closed pattern is a lossless compression of freq. patterns– Reducing the # of patterns and rules
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Closed Patterns and Max-Patterns
• Exercise. DB = {<a1, …, a100>, < a1, …, a50>} – Min_sup = 1.
• What is the set of closed itemset?– <a1, …, a100>: 1
– < a1, …, a50>: 2
• What is the set of max-pattern?– <a1, …, a100>: 1
• What is the set of all patterns?– !!
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Computational Complexity of Frequent Itemset Mining
• How many itemsets are potentially to be generated in the worst case?
– The number of frequent itemsets to be generated is senstive to the minsup threshold
– When minsup is low, there exist potentially an exponential number of frequent itemsets
– The worst case: MN where M: # distinct items, and N: max length of transactions
• The worst case complexty vs. the expected probability
– Ex. Suppose Walmart has 104 kinds of products
• The chance to pick up one product 10-4
• The chance to pick up a particular set of 10 products: ~10-40
• What is the chance this particular set of 10 products to be frequent 103 times in 109 transactions?
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Outline
• Basic Concepts
• The Apriori Algorithm
• The FP-Growth Algorithm
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The Downward Closure Property and Scalable Mining Methods
• The downward closure property of frequent patterns– Any subset of a frequent itemset must be frequent– If {beer, diaper, nuts} is frequent, so is {beer, diaper}– i.e., every transaction having {beer, diaper, nuts} also contains
{beer, diaper} • Scalable mining methods: Three major approaches
– Apriori (Agrawal & Srikant@VLDB’94)– Freq. pattern growth (FPgrowth—Han, Pei & Yin
@SIGMOD’00)– Vertical data format approach (Charm—Zaki & Hsiao
@SDM’02)
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Apriori: A Candidate Generation & Test Approach
• Apriori pruning principle: If there is any itemset which is infrequent, its superset should not be generated/tested! (Agrawal & Srikant @VLDB’94, Mannila, et al. @ KDD’ 94)
• Method:
– Initially, scan DB once to get frequent 1-itemset
– Generate length (k+1) candidate itemsets from length k frequent itemsets
– Test the candidates against DB
– Terminate when no frequent or candidate set can be generated
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The Apriori Algorithm—An Example
Database TDB
1st scan
C1
L1
L2
C2 C2
2nd scan
C3 L33rd scan
Tid Items
10 A, C, D
20 B, C, E
30 A, B, C, E
40 B, E
Itemset sup
{A} 2
{B} 3
{C} 3
{D} 1
{E} 3
Itemset sup
{A} 2
{B} 3
{C} 3
{E} 3
Itemset
{A, B}
{A, C}
{A, E}
{B, C}
{B, E}
{C, E}
Itemset sup{A, B} 1{A, C} 2{A, E} 1{B, C} 2{B, E} 3{C, E} 2
Itemset sup{A, C} 2{B, C} 2{B, E} 3{C, E} 2
Itemset
{B, C, E}
Itemset sup
{B, C, E} 2
Supmin = 2
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The Apriori Algorithm
Ck: Candidate itemset of size kLk : frequent itemset of size k
L1 = {frequent items};for (k = 1; Lk !=; k++) do begin Ck+1 = candidates generated from Lk; for each transaction t in database do
increment the count of all candidates in Ck+1 that are contained in t
Lk+1 = candidates in Ck+1 with min_support endreturn k Lk;
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Implementation of Apriori• How to generate candidates?
– Step 1: self-joining Lk
– Step 2: pruning• Example of Candidate-generation
– L3={abc, abd, acd, ace, bcd}
– Self-joining: L3*L3
• abcd from abc and abd• acde from acd and ace
– Pruning:• acde is removed because ade is not in L3
– C4 = {abcd}
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Further Improvement of the Apriori Method
• Major computational challenges
– Multiple scans of transaction database
– Huge number of candidates
– Tedious workload of support counting for candidates
• Improving Apriori: general ideas
– Reduce passes of transaction database scans
– Shrink number of candidates
– Facilitate support counting of candidates
Partition: Scan Database Only Twice
• Any itemset that is potentially frequent in DB must be frequent in at least one of the partitions of DB– Scan 1: partition database and find local frequent patterns– Scan 2: consolidate global frequent patterns
• A. Savasere, E. Omiecinski and S. Navathe, VLDB’95
DB1 DB2 DBk+ = DB++
sup1(i) < σDB1 sup2(i) < σDB2 supk(i) < σDBk sup(i) < σDB
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DHP: Reduce the Number of Candidates
• A k-itemset whose corresponding hashing bucket count is below the threshold
cannot be frequent
– Candidates: a, b, c, d, e
– Hash entries
• {ab, ad, ae}
• {bd, be, de}
• …
– Frequent 1-itemset: a, b, d, e
– ab is not a candidate 2-itemset if the sum of count of {ab, ad, ae} is below
support threshold
• J. Park, M. Chen, and P. Yu. An effective hash-based algorithm for mining
association rules. SIGMOD’95
count itemsets
35 {ab, ad, ae}
{yz, qs, wt}
88
102
.
.
.
{bd, be, de}...
Hash Table
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Sampling for Frequent Patterns
• Select a sample of original database, mine frequent patterns
within sample using Apriori
• Scan database once to verify frequent itemsets found in
sample, only borders of closure of frequent patterns are
checked
– Example: check abcd instead of ab, ac, …, etc.
• Scan database again to find missed frequent patterns
• H. Toivonen. Sampling large databases for association rules. In
VLDB’96
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DIC: Reduce Number of Scans
ABCD
ABC ABD ACD BCD
AB AC BC AD BD CD
A B C D
{}
Itemset lattice
• Once both A and D are determined frequent, the counting of AD begins
• Once all length-2 subsets of BCD are determined frequent, the counting of BCD begins
Transactions
1-itemsets2-itemsets
…Apriori
1-itemsets2-items
3-itemsDICS. Brin R. Motwani, J. Ullman, and S. Tsur. Dynamic itemset counting and implication rules for market basket data. In SIGMOD’97
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Outline
• Basic Concepts
• The Apriori Algorithm
• The FP-Growth Algorithm
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Pattern-Growth Approach: Mining Frequent Patterns Without Candidate Generation
• Bottlenecks of the Apriori approach
– Breadth-first (i.e., level-wise) search
– Candidate generation and test
• Often generates a huge number of candidates
• The FPGrowth Approach (J. Han, J. Pei, and Y. Yin, SIGMOD’ 00)
– Depth-first search
– Avoid explicit candidate generation
• Major philosophy: Grow long patterns from short ones using local frequent items only
– “abc” is a frequent pattern
– Get all transactions having “abc”, i.e., project DB on abc: DB|abc
– “d” is a local frequent item in DB|abc abcd is a frequent pattern
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Construct FP-tree from a Transaction Database
{}
f:4 c:1
b:1
p:1
b:1c:3
a:3
b:1m:2
p:2 m:1
Header Table
Item frequency head f 4c 4a 3b 3m 3p 3
min_support = 3
TID Items bought (ordered) frequent items100 {f, a, c, d, g, i, m, p} {f, c, a, m, p}200 {a, b, c, f, l, m, o} {f, c, a, b, m}300 {b, f, h, j, o, w} {f, b}400 {b, c, k, s, p} {c, b, p}500 {a, f, c, e, l, p, m, n} {f, c, a, m, p}
1. Scan DB once, find frequent 1-itemset (single item pattern)
2. Sort frequent items in frequency descending order, f-list
3. Scan DB again, construct FP-tree
F-list = f-c-a-b-m-p
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Partition Patterns and Databases
• Frequent patterns can be partitioned into subsets according to f-list– F-list = f-c-a-b-m-p– Patterns containing p– Patterns having m but no p– …– Patterns having c but no a nor b, m, p– Pattern f
• Completeness and non-redundency
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Find Patterns Having P From P-conditional Database
• Starting at the frequent item header table in the FP-tree• Traverse the FP-tree by following the link of each frequent item p• Accumulate all of transformed prefix paths of item p to form p’s
conditional pattern base
Conditional pattern bases
item cond. pattern base
c f:3
a fc:3
b fca:1, f:1, c:1
m fca:2, fcab:1
p fcam:2, cb:1
{}
f:4 c:1
b:1
p:1
b:1c:3
a:3
b:1m:2
p:2 m:1
Header Table
Item frequency head f 4c 4a 3b 3m 3p 3
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From Conditional Pattern-bases to Conditional FP-trees
• For each pattern-base– Accumulate the count for each item in the base– Construct the FP-tree for the frequent items of the
pattern base
m-conditional pattern base:fca:2, fcab:1
{}
f:3
c:3
a:3m-conditional FP-tree
All frequent patterns relate to m
m,
fm, cm, am,
fcm, fam, cam,
fcam
{}
f:4 c:1
b:1
p:1
b:1c:3
a:3
b:1m:2
p:2 m:1
Header TableItem frequency head f 4c 4a 3b 3m 3p 3
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Recursion: Mining Each Conditional FP-tree
{}
f:3
c:3
a:3m-conditional FP-tree
Cond. pattern base of “am”: (fc:3)
{}
f:3
c:3am-conditional FP-tree
Cond. pattern base of “cm”: (f:3){}
f:3
cm-conditional FP-tree
Cond. pattern base of “cam”: (f:3)
{}
f:3
cam-conditional FP-tree
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A Special Case: Single Prefix Path in FP-tree
• Suppose a (conditional) FP-tree T has a shared single prefix-path P
• Mining can be decomposed into two parts
– Reduction of the single prefix path into one node
– Concatenation of the mining results of the two parts
a2:n2
a3:n3
a1:n1
{}
b1:m1C1:k1
C2:k2 C3:k3
b1:m1C1:k1
C2:k2 C3:k3
r1
+a2:n2
a3:n3
a1:n1
{}
r1 =
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Benefits of the FP-tree Structure
• Completeness – Preserve complete information for frequent pattern mining– Never break a long pattern of any transaction
• Compactness– Reduce irrelevant info—infrequent items are gone– Items in frequency descending order: the more frequently
occurring, the more likely to be shared– Never be larger than the original database (not count node-
links and the count field)
30
The Frequent Pattern Growth Mining Method
• Idea: Frequent pattern growth– Recursively grow frequent patterns by pattern and database
partition• Method
– For each frequent item, construct its conditional pattern-base, and then its conditional FP-tree
– Repeat the process on each newly created conditional FP-tree
– Until the resulting FP-tree is empty, or it contains only one path—single path will generate all the combinations of its sub-paths, each of which is a frequent pattern
31
FP-Growth vs. Apriori: Scalability With the Support Threshold
0
10
20
30
40
50
60
70
80
90
100
0 0.5 1 1.5 2 2.5 3
Support threshold(%)
Ru
n t
ime
(se
c.)
D1 FP-grow th runtime
D1 Apriori runtime
Data set T25I20D10K
32
Extension of Pattern Growth Mining Methodology
• Mining closed frequent itemsets and max-patterns– CLOSET (DMKD’00), FPclose, and FPMax (Grahne & Zhu, Fimi’03)
• Mining sequential patterns– PrefixSpan (ICDE’01), CloSpan (SDM’03), BIDE (ICDE’04)
• Mining graph patterns– gSpan (ICDM’02), CloseGraph (KDD’03)
• Constraint-based mining of frequent patterns– Convertible constraints (ICDE’01), gPrune (PAKDD’03)
• Computing iceberg data cubes with complex measures – H-tree, H-cubing, and Star-cubing (SIGMOD’01, VLDB’03)
• Pattern-growth-based Clustering– MaPle (Pei, et al., ICDM’03)
• Pattern-Growth-Based Classification– Mining frequent and discriminative patterns (Cheng, et al, ICDE’07)